Absolutely, those things are all possible.
Some folks install Ignition in a cloud provider and spin up a database on that same provider.
If you’re connecting from an Industrial PC (IPC) to the cloud, you can make a direct database connection (if the firewall rules allow it), but it’s more common and more robust to connect Ignition running on an IPC (often running Ignition Edge) to a cloud Ignition, either via MQTT or over the Gateway Network. In both cases, you can store & forward data if the connection goes down to ensure historical data collection, if that’s what you need.
Hope that helps!